混合Lebesgue空间中平移不变信号的随机采样稳定性

Stability of random sampling for shift-invariant signals in mixed Lebesgue spaces

采样和重构是采样理论中2个重要的问题。在采样集满足稳定性的情况下,信号可以从离散采样集中稳定地重构。为了达到信号重构的目的,对混合Lebesgue空间中平移不变信号的随机采样稳定性进行研究,重点研究能量集中子集上信号的采样稳定性。在生成元满足紧支性和平移稳定性的条件下,定义能量集中子集的标准化子集;研究能量集中子集上信号的无穷范数与(p,q)范数的关系,并借助这一关系估计限制到方体上的标准化子集的覆盖数;根据均匀分布在方体上的随机样本点定义新的随机变量序列并研究其相关性质;借助覆盖数的相关引理、随机变量的相关性质及Bernstein不等式,建立能量集中信号的采样不等式。研究结果表明,当采样点足够多时,能量集中信号的采样稳定性以高概率成立。

Sampling and reconstruction are two important problems in sampling theory.In the case that the set of sampling satisfies the stability,the signal can be stably reconstructed from the discrete sampling set.For the purpose of signal recon-struction,the random sampling stability of shift-invariant signals in mixed Lebesgue spaces is studied,with emphasis on sampling stability of signals on energy concentration subsets.Under the condition that the generator satisfies the support and shift stability,the standardized subset of the energy concentration set is defined firstly.Secondly,we study the relationship between the infinite norm and the(p,q)norm of the signals on the energy concentration subsets and estimate the covering numbers of standardized subsets limited to a cube.Thirdly,the new sequence of random variables is defined according to the random sampling points uniformly distributed on the cube and its related properties are studied.Finally,the sampling ine-quality of energy concentrated signals is established by means of related lemma of covering numbers、corresponding property of random variables and Bernstein inequality.The results show that when there are enough sampling points,the sampling stability of the energy concentrated signals is established with high probability.